Method and system for detecting anomalies in multispectral and hyperspectral imagery employing the normal compositional model

ABSTRACT

A method for detecting anomalies from multidimensional data comprises: a) receiving multidimensional data; b) estimating background parameters of a normal compositional model from the multidimensional data; c) estimating abundance values of the normal compositional model from the background parameters and the multidimensional data; d) determining an anomaly detection statistic from the multidimensional data, the background parameters, and abundance values; and e) classifying an observation from the multidimensional data as either a background reference or an anomaly from the anomaly detection statistic.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.60/394,708, filed 9 Jul. 2002, and is related by common inventorship andsubject matter to the commonly-assigned U.S. Provisional PatentApplication No. 60/394,649 entitled “System and Method for DetectingTargets Known up to a Simplex from Multispectral and HyperspectralImagery Employing the Normal Compositional Model” filed on 9 Jul. 2002.

BACKGROUND OF THE INVENTION

This invention relates generally to image processing systems and moreparticularly to an image anomaly detector for target identification.

Hyperspectral sensors are a new class of optical sensor that collect aspectrum from each point in a scene. They differ from multi-spectralsensors in that the number of bands is much higher (twenty or more), andthe spectral bands are contiguous. For remote sensing applications, theyare typically deployed on either aircraft or satellites. The dataproduct from a hyperspectral sensor is a three-dimensional array or“cube” of data with the width and length of the array corresponding tospatial dimensions and the spectrum of each point as the thirddimension. Hyperspectral sensors have a wide range of remote sensingapplications including: terrain classification, environmentalmonitoring, agricultural monitoring, geological exploration, andsurveillance. They have also been used to create spectral images ofbiological material for the detection of disease and other applications.

With the introduction of sensors capable of high spatial and spectralresolution, there has been an increasing interest in using spectralimagery to detect small objects or features of interest. Anomalydetection algorithms are used to distinguish observations from thebackground when target models are not available or are unreliable.

Anomalies are defined with reference to a model of the background.Background models are developed adaptively using reference data fromeither a local neighborhood of the test pixel or a large section of theimage.

An older method of detecting anomalies from multispectral andhyperspectral imagery is to represent the background imagery usingGaussian mixture models and to use detection statistics derived fromthis model by applying various principles of detection theory. Thisapproach models each datum as a realization of a random vector havingone of several possible multivariate Gaussian distributions. If eachobservation,

y∈R^(n), arises from one of d normal classes then the data have a normalor Gaussian mixture pdf:

$\begin{matrix}{{{p(y)} = {\sum\limits_{k = 1}^{d}\;{\omega_{k}{N\left( {\mu_{k},\Gamma_{k}} \right)}(y)}}},{\omega_{k}\underset{\_}{>}0},{{\sum\limits_{k = 1}^{d}\;\omega_{k}} = 1},} & \left\lbrack {{Eqn}.\mspace{14mu} 1} \right\rbrack\end{matrix}$where ω_(k) is the probability of class k and

${{N\left( {\mu_{k},\Gamma_{k}} \right)}(y)} = {\frac{1}{\left( {2\pi} \right)^{n/2}{\Gamma_{k}}^{1/2}}{\exp\left( {\frac{- 1}{2}\left( {y - \mu_{k}} \right)^{i}{\Gamma_{k}^{- 1}\left( {y - \mu_{k}} \right)}} \right)}}$is the normal probability density function having mean μ_(k) andcovariance Γ_(k). The parameters {(ω_(k), μ_(k), Γ_(k))|1≦k≦d} aretypically estimated from the imagery using defined clusters, theexpectation maximization algorithm or related algorithms such as thestochastic expectation maximization algorithm. Anomaly detection maythen proceed by the application of the generalized likelihood ratio test(GLRT) for an unknown target. Anomaly detection is also accomplished byclassifying each pixel as emanating from one of the d classes—themaximum a posteriori (MAP) principle is one approach toclassification—and applying the GLRT for an unknown target to theclassified data. See, for example, D. W. J. Stein, S. G. Beaven, L. E.Hoff, E. M. Winter, A. P. Schaum, A. D. Stocker, “Anomaly Detection FromHyperspectral Imagery,” IEEE Signal Processing Magazine, January 2001.

Another older approach to anomaly detection is based on the applicationof the linear mixture model. This model accounts for the fact thatpixels in an image are often overlaid with multiple materials so that anobservation may not belong to a class that can be identified with aparticular substance. The linear mixture model represents theobservations, y_(i)∈R^(n), by

$\begin{matrix}{{\left. {{{\left. {{y_{i}\eta} + {\sum\limits_{k = 1}^{d}\;{a_{ki}ɛ_{k}\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu} c{.1}}}} \right)0}\underset{\_}{<}a_{ki}},\mspace{14mu}{{and}\mspace{14mu} c{.2}}} \right){\sum\limits_{k = 1}^{d}\; a_{ki}}} = 1} & \left. {{Eqn}.\mspace{14mu} 2} \right\rbrack\end{matrix}$where, d is the number of classes, ε_(k)∈R^(n), is the signature orendmember of class k, a_(ki) is the abundance of class k in observationy_(i) and η˜N(μ₀, Γ₀) is an additive noise term with normal probabilitydistribution function (pdf) of mean μ₀ and covariance Γ₀. Techniqueshave been developed for estimating the endmembers from the imagery.Given the endmembers, the abundance values are obtained as the solutionto a constrained least squares or a quadratic programming problem.Anomaly detection statistics have been based on the unmixing residual orthe identification of endmembers that represent anomalous classes (seeStein et al. supra).

Spectra from a class of material are often better modeled as randomrather than as fixed vectors. This may be due to biochemical andbiophysical variability of materials in a scene. For such data, neitherthe linear mixture model nor the normal mixture model is adequate, andbetter classification and detection results may accrue from using moreaccurate methods. Stocker et al. [A. D. Stocker and A. P. Schaum,“Application of stochastic mixing models to hyperspectral detectionproblems,” SPIE Proceedings 3071, Algorithms for Multispectral andHyperspectral Imagery III, S. S. Shen and A. E. Iverson eds. August1997.] propose a stochastic mixture model in which each fundamentalclass is identified with a normally distributed random variable, i.e.

$\begin{matrix}{{y_{i} = {{\sum\limits_{k = 1}^{d}\;{a_{ik}ɛ_{k}\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu} ɛ_{k}}} \sim {N\left( {\mu_{k},\Gamma_{k}} \right)}}},{a_{ik}\underset{\_}{>}0},\mspace{14mu}{{{and}{\sum\limits_{k = 1}^{d}\; a_{ik}}} = 1.}} & \left\lbrack {{Eqn}.\mspace{14mu} 3} \right\rbrack\end{matrix}$They estimate the parameters of the model by quantizing the set ofallowed abundance values, and fitting a discrete normal mixture densityto the data. More precisely, let Δ=1/M denote the resolution of thequantization. Then the set of allowed coefficient sequences is

$A = {\left\{ {{\left( {a_{1},\ldots\mspace{14mu},a_{d}} \right)❘{\sum\limits_{j - 1}^{d}\; a_{j}}} = {{1\mspace{14mu}{and}\mspace{14mu} a_{j}} \in \left\{ {0,\Delta,\ldots\mspace{14mu},{\left( {M - 1} \right)\Delta},1} \right\}}} \right\}.}$For each {right arrow over (a)}=(a₁, . . . , a_(d))∈A define

$\begin{matrix}{{\mu\left( \overset{\_}{a} \right)} = {{\sum\limits_{j = 1}^{d}\;{a_{j}\mu_{j}\mspace{14mu}{and}\mspace{14mu}{\Gamma\left( \overset{\_}{a} \right)}}} = {\sum\limits_{j = 1}^{d}\;{a_{j}^{2}{\Gamma_{j}.}}}}} & \left\lbrack {{Eqn}.\mspace{14mu} 4} \right\rbrack\end{matrix}$Then the observations are fit to the mixture model

$\begin{matrix}{{p(y)} = {\sum\limits_{a \in A}\;{\rho_{a}{N\left( {{\mu\left( \overset{\rightharpoonup}{a} \right)},{\Gamma\left( \overset{\rightharpoonup}{a} \right)}} \right)}(y)}}} & \left\lbrack {{Eqn}.\mspace{14mu} 5} \right\rbrack\end{matrix}$

The fitting is accomplished using a variation of the stochasticexpectation maximization algorithm such that Eqn. 4 is satisfied in aleast squares sense. The authors demonstrate improved classification incomparison with clustering methods using three classes, and theydemonstrate detection algorithms using this model. They note, however,that the method is impractical if the data are comprised of a largenumber of classes or if A is small, as the number of elements of A,which is given by:

$\begin{matrix}{{A} = \frac{\left( {M + 1} \right)\ldots\mspace{14mu}\left( {M + d - 1} \right)}{\left( {d - 1} \right)!}} & \left\lbrack {{Eqn}.\mspace{14mu} 6} \right\rbrack\end{matrix}$becomes very large. Furthermore, quantizing the allowed abundance valuesleads to modeling and estimation error.

These unresolved problems and deficiencies are clearly felt in the artand are solved by this invention in the manner described below.

SUMMARY OF THE INVENTION

A method for detecting anomalies from multidimensional data comprises:a) receiving multidimensional data; b) estimating background parametersof a normal compositional model from the multidimensional data; c)estimating abundance values of the normal compositional model from thebackground parameters and the multidimensional data; d) determining ananomaly detection statistic from the multidimensional data, thebackground parameters, and abundance values; and e) classifying anobservation from the multidimensional data as either a backgroundreference or an anomaly from the anomaly detection statistic.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of this invention, reference is nowmade to the following detailed description of the embodiments asillustrated in the accompanying drawing, in which like referencedesignations represent like features throughout the several views andwherein:

FIG. 1 is a block diagram illustrating the Normal Compositional ModelAnomaly Detector embodying various features of the present invention;

FIG. 2 is a block diagram of a flowchart illustrating the parameterestimation method embodying various features of the present invention;and

FIG. 3 is a block diagram illustrating the method of this invention forinitializing the endmember classes.

DESCRIPTION OF THE PREFERRED EMBODIMENT

The invention is used to detect anomalies in spectral imagery. Anembodiment of the invention is operated as shown in FIGS. 1–3. As shownin FIG. 1, the embodiment has four major components: estimation of theparameters of the normal compositional model (NCM), estimation of theabundance values of the classes of the NCM, computation of the detectionstatistic, and application of a decision criterion to distinguishanomalies from background based on the values of the detectionstatistic. These steps are elucidated below.

1. Normal Compositional Model

The normal compositional model represents each observation y_(i)∈R^(n)as:

$\begin{matrix}{{{\left. {{\left. {{{\left. {y_{i} = {{c\;\eta} + {\sum\limits_{k - 1}^{d}\;{a_{ki}ɛ_{k}\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu} c{.1}}}}} \right)0}\underset{\_}{<}a_{ki}},\mspace{14mu}{{and}\mspace{14mu} c{{.2}.a}}} \right){\sum\limits_{k - 1}^{d}\; a_{ki}}} = {1\mspace{14mu}{or}\mspace{14mu} c{{.2}.b}}} \right){\sum\limits_{k - 1}^{d}\; a_{ki}}}\underset{\_}{<}1},} & \left\lbrack {{Eqn}.\mspace{14mu} 7} \right\rbrack\end{matrix}$where ε_(k),η∈R^(n) are random vectors such that ε_(k)˜N(μ_(k),Γ_(k)),η˜N(μ₀,Γ₀) and c=0,1. Constraint c.2.b may be used in place of c.2.a toaccount for variations in scale or as in remote sensing, scalarvariations in illumination. Applied to remote sensing data, η modelspath radiance, additive sensor noise, and other additive terms. Bychoosing c=0, and constraints c.1 and c.2.a, the model reduces to theSchaum-Stocker model (Eqn. 3). Although, with this choice of parametersand constraints this embodiment has advantages over the Schaum-Stockerapproach since the estimation procedure does not confine the abundancevalues to preselected quantized values. Therefore, it is not restrictedto a small number of classes and it provides more accurate estimates ofclass parameter and abundance values. This model reduces to the linearmixing model by choosing Γ_(k)=0 for all 1≦k≦d and c=1, although theparameter estimation technique described below will not refine initialestimates of the μ_(k) in this case. It does, however, provide a maximumlikelihood approach to estimating the parameters of the distribution ofη. Furthermore, by imposing the constraints c.2.a and a_(ki)=0,1 foreach 1≦i≦N, exactly one of a_(ki)=1, and the model encompasses theGaussian mixture model. Whereas specialized constraints applied to theparameters of the NCM reduce it to the older models, in general, withoutimposing special constraints, the NCM provides a model having higherlikelihood than these alternatives.2. Parameter Estimation

The parameter estimation module is illustrated in FIG. 2 and describedbelow.

A. Initialization

The initialization module is depicted in FIG. 3. The mean value of theadditive term η is obtained as a virtual shade point, and the covarianceof the additive term is estimated as the sample covariance of a clusterof points near η. Initial estimates of the class means are obtained byapplying deterministic linear unmixing techniques to determine a set ofendmembers. Several methods of estimating endmembers are available.Clusters containing a prescribed number of points near each endmemberare defined, and an initial estimate of each class covariance isobtained as the sample covariance of the corresponding cluster.

B. Updating Abundance Estimates (UA)

For given parameters

(μ_(k), Γ_(k)), 1≦k≦d, and given abundances α_(i)=(a_(1i), . . .,a_(di)), let

$\begin{matrix}{{\mu\left( \alpha_{i} \right)} = {\sum\limits_{k = 1}^{d}\;{a_{ki}\mu_{k}}}} & \left\lbrack {{Eqn}.\mspace{14mu} 8} \right\rbrack\end{matrix}$and

$\begin{matrix}{{\Gamma\left( \alpha_{i} \right)} = {\sum\limits_{k = 1}^{d}\;{\left( a_{ki} \right)^{2}{\Gamma_{k}.}}}} & \left\lbrack {{Eqn}.\mspace{14mu} 9} \right\rbrack\end{matrix}$

Then, y_(i)˜N(μ(α_(i))+μ₀,Γ(α_(i))+η₀). Maximum likelihood abundanceestimates are thus obtained by solving

$\begin{matrix}{{\hat{\alpha}}_{i} = {\arg\left( {\max\limits_{\alpha_{i}}\left( {\frac{1}{{{{\Gamma\left( \alpha_{i} \right)} + \Gamma_{0}}}^{05}\left( {2\pi} \right)^{n\text{/}2}}{\quad{\exp\left( {\frac{- 1}{2}\left( {y_{i} - {\mu\left( \alpha_{i} \right)} - \mu_{0}} \right)^{i}\left( {{\Gamma\left( \alpha_{i} \right)} + \Gamma_{0}} \right)^{- 1}\left( {y_{i} - {\mu\left( \alpha_{i} \right)} - \mu_{0}} \right)} \right)}}} \right.} \right.}} & \left\lbrack {{Eqn}.\mspace{14mu} 10} \right\rbrack\end{matrix}$subject to the constraints c.1, and c.2.a, or c.2.b.

C. Update Class Parameters (UP)

For given abundance estimates, the class parameters,Ω={(μ_(k),Γ_(k))|0≦k≦d}, may be estimated by applying theexpectation-maximization (EM) algorithm. Let

$\Omega^{r} = \left\{ {\left( {\mu_{k}^{r},\Gamma_{k}^{r}} \right)❘{0\underset{\_}{<}k\underset{\_}{<}d}} \right\}$denote the estimate of the parameters after the r^(th) iteration of theEM algorithm. Given the abundance values {a_(ki)|1≦i≦N,1≦k≦d}, define

${\delta_{i}^{r} = {\left\lbrack {{\Gamma^{r}\left( \alpha_{i} \right)} + {c\;\Gamma_{0}^{r}}} \right\rbrack^{- 1}\left( {y_{i} - {\mu^{r}\left( \alpha_{i} \right)} - {c\;\mu_{0}^{r}}} \right)}},\mspace{14mu}{\psi_{ki}^{r} = {\alpha_{ki}\delta_{i}^{r}}},{{\overset{\_}{\delta}}^{r} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\;\delta_{i}^{r}}}},\mspace{14mu}{{{and}\mspace{14mu}{\overset{\_}{\psi}}_{k}^{r}} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\;{\psi_{ki}^{r}.}}}}$The EM update equations are:μ_(k) ^(r+1)=μ_(k) ^(r)+Γ_(k) ^(r) ψ _(k) ^(r), for 1≦k≦d.  [Eqn. 11a]μ₀ ^(r+1)=μ₀ ^(r)+Γ₀ ^(r) δ ^(r).(c=1)  [Eqn. 11b]

$\begin{matrix}{{\Gamma_{k}^{r + 1} = {\Gamma_{k}^{r} - {{\Gamma_{k}^{r}\left( {\frac{1}{N}{\sum\limits_{i = 1}^{N}\;{\alpha_{ki}^{2}\left\lbrack {{\Gamma^{r}\left( \alpha_{i} \right)} + {c\;\Gamma_{0}^{r}}} \right\rbrack}^{- 1}}} \right)}\Gamma_{k}^{r}} + {{\Gamma_{k}^{r}\left( {{\frac{1}{N}{\sum\limits_{i = 1}^{N}\;{\psi_{ki}^{r}\psi_{ki}^{r^{T}}}}} - {{\overset{\_}{\psi}}_{k}^{r}{\overset{\_}{\psi}}_{k}^{r^{T}}}} \right)}\Gamma_{k}^{r}}}},\mspace{14mu}{{{for}1}\underset{\_}{<}k\underset{\_}{<}{d.}}} & \left\lbrack {{{Eqn}.\mspace{14mu} 11}c} \right\rbrack\end{matrix}$

$\begin{matrix}{\Gamma_{0}^{r + 1} = {\Gamma_{0}^{r} - {\frac{1}{N}\Gamma_{0}^{r}{\sum\limits_{i = 1}^{N}\;{{c^{2}\left\lbrack {{\Gamma^{r}\left( \alpha_{i} \right)} + {c\;\Gamma_{0}^{r}}} \right\rbrack}^{- 1}\Gamma_{0}^{r}}}} + {{\Gamma_{0}^{r}\left( {{\frac{1}{N}{\sum\limits_{i = 1}^{N}\;{\delta_{i}^{r}\delta_{i}^{r^{T}}}}} - {{\overset{\_}{\delta}}^{r}{\overset{\_}{\delta}}^{r^{T}}}} \right)}{\Gamma_{0}^{r} \cdot \left( {c - 1} \right)}}}} & \left\lbrack {{{Eqn}.\mspace{14mu} 11}d} \right\rbrack\end{matrix}$The class parameters are updated (UP) using the expectation-maximizationequations (Eqns. 11) and the current abundance estimates

{a_(ik)^(i)}.Likelihood increases with each iteration of UA or UP. Thus, a sequenceof parameter estimates of increasing likelihood is obtained by theapplication of a sequence of updates: UA, UP, UA, UP, . . . . Theiteration is halted when a convergence criterion is satisfied.3. Detection Algorithms

Anomaly detection algorithms may be derived from the NCM. An anomalydetection statistic is obtained by estimating the parameters of the dataas described above and computing the log-likelihood of the observation,y_(i), given the parameters:

$\begin{matrix}{{A_{NC}\left( y_{i} \right)} = {{- {L\left( {y_{i}❘H_{0}} \right)}} = {{{- \frac{1}{2}}{\log\left( {{{\Gamma\left( \alpha_{i} \right)} + \Gamma_{0}}} \right)}} - {\frac{n}{2}{\log\left( {2\pi} \right)}} - {\frac{1}{2}\left( {y_{i} - {\mu\left( \alpha_{i} \right)} - \mu_{0}} \right)^{t}\left( {{\Gamma\left( \alpha_{i} \right)} + \Gamma_{0}} \right)^{- 1}{\left( {y^{i} - {\mu\left( \alpha_{i} \right)} - \mu_{0}} \right).}}}}} & \left\lbrack {{Eqn}.\mspace{14mu} 12} \right\rbrack\end{matrix}$

An anomaly detection procedure is obtained by comparing (12) to athreshold that is obtained from background data.

The class parameters may be updated using a segmented expectationmaximization algorithm in place of the expectation-maximizationalgorithm. In this approach a lower threshold, possibly zero, is placedon the abundance of a class, and only those pixels for which theabundance exceeds the threshold are utilized in the update of theassociated class parameters. This approach saves computations andimproves the speed of convergence of the parameter estimates.

Rather than solving for the maximum likelihood value of the abundanceestimates in the parameter estimation phase of the operation, randomsamples of the abundance estimates may be generated and these may beused in place of the maximum likelihood estimates in the updating of theclass parameters.

Referring to FIG. 1, there is shown a block diagram of an embodiment ofa process 100 for detecting anomalies from multidimensional data. 112Multidimensional data 112 is received at step 120 which estimatesbackground parameters 122 of a normal compositional model from themultidimensional data 112. At step 130, abundance values 132 of thenormal compositional model are estimated from the background parameters122 and multidimensional data 112. Continuing to step 140, an anomalydetection statistic 142 is determined from the multidimensional data112, background parameters 122, and abundance values 132. At step 150,an observation 152 classification is determined from themultidimensional data 112 as either a background reference or an anomalyfrom the anomaly detection statistic 142.

FIG. 2 illustrates that step 120, for estimating background parameters,122 further includes initializing current class parameters 222 andcurrent abundance estimates 224 at step 220, where such current classparameters 222 and current abundance estimates 224 are collectivelyreferenced as signal 226. Next, at step 230, updated abundance estimates232 are defined from the current abundance estimates 224, current classparameters 222, and multidimensional data 112. Step 240 determinesconverged class parameter candidates 242 from the updated abundanceestimates 232, current class parameters 226, and multidimensional data112. At step 270, a first affirmative convergence signal 122 isgenerated if the converged class parameter candidates 242 satisfy firstconvergence criteria, or step 270 generates a non-convergence signal 274that is provided to step 230, whereupon step 120 returns to step 230 ifthe converged class parameter candidates 242 do not satisfy the firstconvergence criteria.

As shown in FIG. 2, step 240 further includes creating updated currentclass parameters 252 at step 250 from the current class parameters 222,current abundance estimates 224, and multidimensional data 112.Converged class parameters candidates 242 are provided from step 260 tostep 270 if updated current class parameters 252 satisfy secondconvergence criteria. Step 260 generates a non-convergence signal 264that is provided to step 250, whereupon step 240 returns to step 250 ifupdated current class parameters 242 do not satisfy the secondconvergence criteria.

Referring to FIG. 3, the process of initializing the current classparameters 226 includes: 1) defining a shade point offset value 312 atstep 310; 2) defining a shade point covariance value 322 at step 320from the shade point offset value 312 and multidimensional data 112; 3)defining end members 332 from the shade point value 312 andmultidimensional data 112 at step 330; and 4) at step 340, initializinga covariance matrix 226 that is a composite of signals 222 and 224 foreach of the end members 332 from the multidimensional data 112.

Clearly, other embodiments and modifications of this invention may occurreadily to those of ordinary skill in the art in view of theseteachings. Therefore, this invention is to be limited only by thefollowing claims, which include all such embodiments and modificationswhen viewed in conjunction with the above specification and accompanyingdrawing.

1. A method for detecting anomalies from multidimensional data,comprising: a) receiving multidimensional data; b) estimating backgroundparameters of a normal compositional model from said multidimensionaldata; c) estimating abundance values of said normal compositional modelfrom said background parameters and said multidimensional data; d)determining an anomaly detection statistic from said multidimensionaldata, said background parameters, and said abundance values; and e)classifying an observation from said multidimensional data as either abackground reference or an anomaly from said anomaly detectionstatistic.
 2. The method of claim 1 wherein estimating backgroundparameters further includes: f) initializing current class parametersand current abundance estimates from said multidimensional data; g)defining updated abundance estimates from said current abundanceestimates, said current class parameters, and multidimensional data h)determining converged class parameter candidates from said updatedabundance estimates, said current class parameters, and saidmultidimensional data; and i) generating said background parameters ifsaid converged class parameter candidates satisfy first convergencecriteria or returning to said defining updated abundance estimates ifsaid converged class parameter candidates do not satisfy said firstconvergence criteria.
 3. The method of claim 2 wherein determining saidconverged class parameter candidates further includes: j) creatingupdated current class parameters from said current class parameters,said current abundance estimates, and said multidimensional data; and k)generating said converged class parameter candidates from said updatedcurrent class parameters if said updated current class parameterssatisfy second convergence criteria, or returning to said creatingupdated current class parameters if said updated current classparameters do not satisfy said second convergence criteria.
 4. Themethod of claim 2 wherein said initializing said current classparameters includes: l) defining a shade point offset value from saidmultidimensional data; m) defining a shade point covariance value fromsaid shade point offset value and said multidimensional data; n)defining end members from said shade point value and saidmultidimensional data; and o) initializing said current class parametersand said current abundance estimates from said end members and saidmultidimensional data.
 5. The method of claim 1 wherein saidmultidimensional data is detected by an imaging spectrometer.
 6. Themethod of claim 1 wherein said multidimensional data represents surfacespectra.